Magnetism is a spatial relativity spell

There are many explanations of magnetism in one reference frame turning into a electric force in another, using spacial relativity that talks about physical properties (distance, mass and time) changes with velocity.

So magnetic force is just an illusion in our inertial reference. In another velocity reference where the moving charge stops, the magnetic force can be interely converted in a pure electrostatic force, due to shrinkage or expansion of lenght and time, related to relative charge positions.

I have always found this amazing, because it is a daily manifestation of our everyday reality of a part of Einstein’s Theory of Relativity

However almost all cases are relative to a case of a point charge moving parallel to a wire with electric current (for instance, here)

In this figure, green vector is a resulting force, yellow vector is magnetic field (the cross indicates that the vector is crossing the screen away from us) and purple vector is velocity. Light green vector is the repulsion force from protons in the wire and light red vector is the attractive force from drift electrons in the wire.

It’s easy to understand that when we consider the reference frame of point charge, if we suppose that the velocity is equal to the drift velocity of electrons in the wire that is the net velocity of electric current.

The negative charges become denser because its relative velocity turns to 0, once we are in a frame of reference in same drift velocity of electrons. The positive charges become more scattered with Lorentz effect of special relativity because protons start to move relatively in the opposite direction with drift velocity.

So the repulsive force of protons (considering in same number of drift electrons) exceeds the attractive force of eletrons.

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I don’t find in Internet any reference to other velocity directions of point charges related to a wire with electric current. The situation showed is always the same (point charge movement parallel to wire with electric current)

I’ve found the article “The simplest, and the full derivation of Magnetism
as a Relativistic side effect of ElectroStatics” by Hans de Vries (2008). He talks about perpendicular case, but in a very formal way, without no intuitive understanding.

I also have found the article “The Correct Derivation of Magnetism from Electrostatics Based on Covariant Formulation of Coulomb’s Law” by
Mueiz Gafer KamalEldeen, self-appointed independent researcher. This article tries to address the question of the charge that travels perpendicular to a wire with electric current.

Unfortunately, this text is not trustable and contains errors and one of them is particularly serious and fool, when the author shows a figure the produces a force that violates the right hand rule of magnetic Lorentz Force.

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The below explanation is not a proof, it’s more like a visual intuition, because it seems so weird for students understand magnetism just with dry math. It’s seems like wizardry.

Everything starts with an image:

We want to show that a charge point approaching a wire with electric current in normal direction, can result in a force to the left. In the case of the figure, the electric current flows to the right and, therefore, the electron drift goes to the left. The figure show that the point charge is positive.

The gray retangle is a wire with electric current, velocity is a violet vector, force is a green vector and magnetic field is yellow vector, indicated by X because the . The magnetic field crosses the page moving away from us.

The blue vectors indicate attractive forces (electrons) and pink vectors are repulsive forces. R is a force from some charge particle in the right side e L is the same to the left.

The blue and mustard values are calculated with Pitagoras and show the distance between a point charge and some hypotetical drift electron or some corresponding proton. There are 2 different moments: The original (mustard) and new (blue) position, after a time lapse.

The values in red indicates the absolute difference between above distances. The bigger values are near normal line.

If the velocity is perpendicular to the wire, let’s look at the drift electrons and the protons in the same number on the opposite direction, because the rest of the particles (electrons and protons), have a very fast random motion, without predominant direction, so it results in equal forces in the modulus and opposite in the direction.

Here we will put in the reference of the point charge. Then at the blue dot above the charge, it is as if the wire go down that same displacement instead of the charge going up.

We do not represent the wire down movement to make the figure clearer.

On the right side, it is easy to see that the electrons are approaching to normal direction to point charge by drift velocity.

The figure shows that after a lapse of time, the absolute value of displacement increases as it approaches the normal to the point. This represents a greater speed and therefore a greater distance shrinking by Lorentz effect.

This shrinking increases the electrostatic attractive force of the near normal electrons and deflects the resulting direction more to vertical (R-). This situation isn’t so strong with protons in the same number of the drift electrons, represented by R+.

Thus R- is more vertical than R+.

On the left side, in turn, it is easy to see that the electrons move away from the normal by the drift. Thus the drift to the left, considering a lapse of time, causes the relative velocity to decrease, which lengthens compared to the absence of drift, and therefore decreases the electrostatic attractive force and deflects the resulting direction more to horizonal (L-).

This situation is weaker with protons in the same number of drift electrons, represented by L+. Therefore, L- is more horizontal of L+, which has the same angle to the vertical, by symmetry, to R+.

Only with precise calculation we can prove, but it is possible to imagine that the largest vector near the vertical direction (R-) plus the smallest vector near the horizontal direction (L-), both upwards, are balanced with the average vectors of the positive charges of the wire that points downwards.

Likewise, it is very likely that the horizontal component of the vector L- to the left more than compensates the horizontal R- component. This is possible because at relatively low angles the cosine approaches much more than 1 than the sine approaches 0. For example, for an angle of 20°. this difference has an approximate factor of 5.7!

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Intuition is great teaching tool to undestanding our world and the relativist interpretations of magnetic phenomenon is treated in an incomplete manner from a more visual point of view, generally only addressing the situation with the velocity of point charge parallel to the wire.

For me it’s clear that the association of electrons drift and point charge movement parallel to magnetic field has a kind of symmetry that stops any drift effect in order to produce a force.

So, in 2 pures dimensions (velocity perpendicular and parellel to the wire), the force is perpendicular to velocity following the right hand rule, however if velocity is parallel to magnetic field, there is no generated force.

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For more math-oriented guys:

With a scent of cross product, we can imagine a new vector field B⃗ and call it magnetic field that is proportional to generated force.

What dependences?
Just the distance from the wire (supposed in the air). And the electric current of course.

Is is constant around the wire?
Yes, by symmetry.

There is some radial component to or away from the wire?
No, because two observers looking at the wire in opposite directions notice equal magnetic field direction but inverted current orientations. It does not happen with a vector tangent to a imaginary circle around the wire. While one viewer sees B⃗ clockwise the other observer sees B⃗ counterclockwise.

What value?

It should match the calculated value here using special relativity:

F = q v ( μ₀ / 2π r)

Where r is the distance between charge and wire. So it’s convenient consider the B⃗ module as:

|B⃗| = μ₀ / 2π r

Thus magnetic field points to tangent in a imaginary circle around the wire in the direction that matches with the following cross product.

F ⃗ = qv⃗ × B⃗

Where B⃗ is the magnetic field with direction and orientation from right hand rule, v⃗ is a velocity of target body and F⃗ is a resultant force.

So, as seen above, where v⃗ and B⃗ are parallel there is no force.

In general, F⃗ would need to be perpendicular to v⃗ . Suppose v⃗ had a component parallel to force and put a moving charge in a constant magnetic field: the charge will accelerate forever creating energy from nothing.

This magnetic field value matches with Ampere-Maxwell Law applied to a wire with constant electric current.

∮ᶜ B⃗ dr = μ₀ I

Where I is the oriented current passing through a surface delimited by the closed curve of the integral. Thus:

B 2π r = μ₀ I
B = μ₀ I / 2πr